The Black Hole Information Loss Problem

In 1975 Hawking and Bekenstein made a remarkable connection between thermodynamics,
quantum mechanics and black holes, which predicted that black holes will slowly radiate
away. (see Relativity FAQ Hawking Radiation). It
was soon realized that this prediction created an information loss problem that has since
become an important issue in quantum gravity.

In order to understand why the information loss problem is a problem, we need
first to understand what it is. Take a quantum system in a pure state and throw it
into a black hole. Wait for some amount of time until the hole has evaporated enough
to return to its mass previous to throwing anything in. What we start with is a pure
state and a black hole of mass M. What we end up with is a thermal state and a black
hole of mass M. We have found a process (apparently) that converts a pure state
into a thermal state. But, and here's the kicker, a thermal state is a MIXED state
(described quantum mechanically by a density matrix rather than a wave function). In
transforming between a mixed state and a pure state, one must throw away
information. For instance, in our example we took a state described by a set of
eigenvalues and coefficients, a large set of numbers, and transformed it into a state
described by temperature, one number. All the other structure of the state was lost
in the transformation.

In technical jargon, the black hole has performed a non-unitary transformation on the
state of system. As you may recall, non-unitary evolution is not allowed to occur
naturally in a quantum theory because it fails to preserve probability; that is, after
non-unitary evolution, the sum of the probabilities of all possible outcomes of an
experiment may be greater or less than 1.

In the face of such evolution, quantum mechanics falls apart, and we are faced with a
dilemma. Do black holes really defy the tenets of quantum theory, or have we missed
something in our thought experiment. Perhaps the black hole is not the same after it
has evaporated to mass M as it was initially at mass M. Or perhaps there is some
subtle correlation in the Hawking radiation that we are missing, but that supplies the
missing information about the pure state.

This, then, is the black hole information loss problem. The fact that information
is lost is reflected in the thermal nature of the emitted radiation. But any thermal
system can be assigned an entropy via the Gibbs law dE = S dT. Thus, we can
calculate the black hole entropy by dint of the fact that we can calculate the black hole
temperature (by dint of the fact that the quantum radiation is thermal). This is, I
think, what people are getting at when they say that black hole entropy is responsible for
the information loss. I would say it the other way, that black hole information loss
is responsible for black hole entropy. The simple fact of the matter is that they
are the same thing in slightly different terms.

Two notes to finish off. First, you might think that the thermal nature of the
black hole is inevitable since it is radiating, but you would be wrong. In most of
these quantum radiation calculations, the spectrum of the radiation does not have a Planck
spectrum. If that had been the case for black holes, too, then we would not be able
to assign a temperature or an entropy to black holes. In that case, people probably
still would not believe Bekenstein and instead of the information loss paradox we'd still
be wondering how to reconcile black holes with the second law. The thermal spectrum
of Hawking radiation is one of the most serendipitous results in modern physics, in my
opinion, which is another way of saying that something deep and not understood is going
on.

The second is an interesting sidelight. While it's true that the Gibbs law gives
the correct Bekenstein-Hawking entropy from the calculated temperature, no one has been
able (until a few months ago) to explain the entropy directly from quantum mechanical /
statistical mechanical grounds. In fact, it has been proven that semiclassical
gravity is insufficient to account for this entropy. This is a profound result,
since the thermodynamical entropy is obtained at a semiclassical level (in fact, due to
some quirks that I suspect are related to the non-linearity of gravity, it is essentially
classical). Thus, we are faced with the disconcerting choice that A) thermodynamical
entropy does not always have a statistical mechanical basis or B) gravity is not a
fundamental interaction, but rather a composite effect of some more fundamental underlying
theory. Option B is not disconcerting to superstring theorists, however, it is
exactly their point of view. Interestingly, since about the beginning of the year, the
superstring people have jumped into the "origin of black hole entropy" fray. It
turns out that by using some old result about monopoles in certain types of field theories
they have been able to count the string states that would contribute to a certain
(unphysical) class of a black hole of a given mass. The entropy is exactly that
given by the Bekenstein area formula. The experts assure me that this will be
extended to more physical models in the future. An exciting prospect indeed, if it
pans out.